Asymptotics for configural location estimators
type: Item
instance of: scholarly article
title: Asymptotics for configural location estimators
zbMATH Open document ID: 0603.62031
author: Stephan Morgenthaler
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: Let \(x_ 1,...,x_ n\) be independent real-valued observations from the distribution function (d.f.) F(x-\(\mu)\); it is assumed that F(\(\cdot)\) is symmetric around 0, \(0<F(x)<1\) for all \(x\in {\mathbb{R}}\), and has a probability density function f(\(\cdot)\) with respect to Lebesgue measure. When sampling from a symmetric situation as above, the Pitman estimator of \(\mu\) is given by \[ T_ F(x_ 1,...,x_ n)=\int^{\infty}_{- \infty}r\prod^{n}_{i=1}f(x_ i+r)dr/\int^{\infty}_{- \infty}\prod^{n}_{i=1}f(x_ i+r)dr. \] This estimator has the smallest mean-square-error among all location equivariant estimators, where T(\(\cdot)\) is location equivariant, if \(T(x_ 1+r,...,x_ n+r)=T(x_ 1,...,x_ n)+r\), \(r\in {\mathbb{R}}.\) In this paper, the main interest is centered at the behavior of the so- called bioptimal or compromise location estimators when the sample is coming from a d.f. G(\(\cdot)\) symmetric around 0. With d.f.'s \(F_ 1(\cdot)\) and \(F_ 2(\cdot)\) satisfying the same assumptions as the d.f. F(\(\cdot)\) above, a compromise estimator \(T_{F_ 1,F_ 2,\alpha}(x_ 1,...,x_ n)\) is defined by \[ T_{F_ 1,F_ 2,\alpha}(x_ 1,...,x_ n)= \] \[ \alpha w_{F_ 1}(x_ 1,...,x_ n)T_{F_ 1}(x_ 1,...,x_ n)+(1-\alpha)w_{F_ 2}(x_ 1,...,x_ n)T_ F(x_ 1,...,x_ n), \] where \(0<\alpha <1\); for \(k=1,2\), \(T_{F_ k}(x_ 1,...,x_ n)\) are given as \(T_ F(x_ 1,...,x_ n)\) above, and the weights \(w_{F_ k}(x_ 1,...,x_ n)\) are given by \[ w_{F_ k}(x_ 1,...,x_ n)= \] \[ \int^{\infty}_{- \infty}\prod^{n}_{i=1}f_ k(x_ i+r)dr/\int^{\infty}_{- \infty}\{\alpha \prod^{n}_{i=1}f_ 1(x_ i+r)+(1- \alpha)\prod^{n}_{i=1}f_ 2(x_ i+r)\}dr, \] k\(=1,2\). In many applications, \(F_ 1(\cdot)\) is chosen to be of the Gaussian shape, and, on the basis of various considerations, \(F_ 2(\cdot)\) is chosen as a heavy tailed distribution; for example, a distribution with Pareto tails. Under certain overly nice conditions, which do not include, for example, distributions with Pareto tails, the asymptotic behavi…
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62G20
zbMATH DE Number: 3974060
zbMATH Keywords: configural polysampling
zbMATH Keywords: symmetric distribution
zbMATH Keywords: Pitman estimator
zbMATH Keywords: mean- square-error
zbMATH Keywords: location equivariant estimators
zbMATH Keywords: compromise estimator
zbMATH Keywords: heavy tailed distribution
zbMATH Keywords: Pareto
zbMATH Keywords: asymptotic behavior of the weights
zbMATH Keywords: asymptotic variances
zbMATH Keywords: standard normal distribution
zbMATH Keywords: Huber's minimax estimator
reviewed by: George Roussas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349848
OpenAlex ID: W2055506776
DOI: 10.1214/AOS/1176349848
description: scientific article; zbMATH DE number 3974060